Star Gravitational Binding Energy Questions

[Assaltwaffle]

Hello everyone! I am a moderator over at the VS Battles Wiki, and I have some questions I believe one of y'all can answer. Basically the question is: how much energy does it take to destroy a star? We find this with GBE, but I have some questions as to calculating this value.

As far as I am aware, we can calculate GBE using the formula U = (3*G*M^2)/(5*r), but it seems like this formula gives results that are different than what some sources say the Sun's GBE is. Upfront, plugging in the values we know of the Sun into the formula, U = 2.277164x10^41 seems to be the answer. However, in these sources the Sun's GBE is cited to be 3.8x10^41. Still in the same ballpark, but not the same, and different from what is calculated. For the Stellar Structure source, scroll down to page 5.

Another user pointed out that stars have far more dense cores than outer layers, and thus the GBE formula lowballs their true GBE. However I question this, as attempting to calculate the Earth's GBE using the aforementioned formula still gives a result very close to a well-sourced number, despite the Earth also being more centrally dense due to our iron core. He then pointed to https://ia800602.us.archive.org/26/items/AnIntroductionToTheStudyOfStellarStructure/Chandrasekhar-AnIntroductionToTheStudyOfStellarStructure_text.pdf (page 101, equation 90), saying that n needs to be added to the formula to compensate for non-uniform density. However it seems like this is in reference to Omega, which was previously used as Gravitational Potential Energy.

Is GPE the same as GBE?

So my big question is this: Does average density account for the increased core density when using the aforementioned GBE formula, or does it not compensate enough? If the formula does not account for non-uniform density enough and an additional variable is needed, and that variable is n of the formula listed on page 101, how do we find n for other stars?

As it stands stars such as UY Scuti are less durable than our own Sun, given the formula, which doesn't seem to make sense, since large stars have more dense cores with higher thermal energy than our Sun, or so I have been told by the aforementioned user. In fact, according to the information on radius and mass on many stars, smaller stars seem to be more durable than larger stars, and vice versa. This seems to be very unusual, as I would assume that a star measuring several AU in radius would be higher in GBE than a star smaller than our own Sun.

Here is the full thread for those interested (and dedicated). I really appreciate any help I can get, as it seems like for every answer I find I come out with another question. Thanks for your time!http:// https://www.astro.umd.edu/~jph/A320_Stellar_Structure.pdf

 

[PeterDonis]

stars have far more dense cores than outer layers, and thus the GBE formula lowballs their true GBE. However I question this, as attempting to calculate the Earth's GBE using the aforementioned formula still gives a result very close to a well-sourced number, despite the Earth also being more centrally dense due to our iron core

The variation in the Earth's density is much less than the variation in the Sun's density. The Earth's density varies over less than an order of magnitude, from about 3 g/cm^3 at the surface to about 13 g/cm^3 in the core; its average density is about 5.5 g/cm^3. The Sun's density varies from less than 1 gm/cm^3 near the surface (possibly much less, depending on how you define the "surface"), to about 160 gm/cm^3 in the core, or well over 2 orders of magnitude; its average density is about 1.4 gm/cm^3.

This means the GBE formula you are using is much more of an underestimate for the Sun than for the Earth.

Is GPE the same as GBE?

As the term is being used in the source you referred to, yes.

Does average density account for the increased core density when using the aforementioned GBE formula, or does it not compensate enough?

As you found for the Sun, it does not compensate enough.

If the formula does not account for non-uniform density enough and an additional variable is needed, and that variable is n of the formula listed on page 101, how do we find n for other stars?

The polytrope model (i.e., the model where you have a number $n$ that describes how the density varies in the object--btw, $n$ is a constant for a given star in this model) is only an approximation, but it's a fairly good one for many stars. This Wikipedia article gives some typical values of $n$ for different types of objects:

https://en.wikipedia.org/wiki/Polytrope

 

[Assaltwaffle]

The variation in the Earth's density is much less than the variation in the Sun's density. The Earth's density varies over less than an order of magnitude, from about 3 g/cm^3 at the surface to about 13 g/cm^3 in the core; its average density is about 5.5 g/cm^3. The Sun's density varies from less than 1 gm/cm^3 near the surface (possibly much less, depending on how you define the "surface"), to about 160 gm/cm^3 in the core, or well over 2 orders of magnitude; its average density is about 1.4 gm/cm^3.

This means the GBE formula you are using is much more of an underestimate for the Sun than for the Earth.
As the term is being used in the source you referred to, yes.
As you found for the Sun, it does not compensate enough.
The polytrope model (i.e., the model where you have a number $n$ that describes how the density varies in the object--btw, $n$ is a constant for a given star in this model) is only an approximation, but it's a fairly good one for many stars. This Wikipedia article gives some typical values of $n$ for different types of objects:

https://en.wikipedia.org/wiki/Polytrope

Very insightful. Thank you so much for your contributions! This should very much help in putting a more accurate number for our star levels. This was such a quick response, as well. I feel very welcomed to ask questions here; thanks for that.

 

[Assaltwaffle]

The variation in the Earth's density is much less than the variation in the Sun's density. The Earth's density varies over less than an order of magnitude, from about 3 g/cm^3 at the surface to about 13 g/cm^3 in the core; its average density is about 5.5 g/cm^3. The Sun's density varies from less than 1 gm/cm^3 near the surface (possibly much less, depending on how you define the "surface"), to about 160 gm/cm^3 in the core, or well over 2 orders of magnitude; its average density is about 1.4 gm/cm^3.

This means the GBE formula you are using is much more of an underestimate for the Sun than for the Earth.
As the term is being used in the source you referred to, yes.
As you found for the Sun, it does not compensate enough.
The polytrope model (i.e., the model where you have a number $n$ that describes how the density varies in the object--btw, $n$ is a constant for a given star in this model) is only an approximation, but it's a fairly good one for many stars. This Wikipedia article gives some typical values of $n$ for different types of objects:

https://en.wikipedia.org/wiki/Polytrope

Also, seeing as red giants have a lower Polytrope value than main-sequence stars, wouldn't this indicate that the Giants still have a lower GBE than the Sun and Carbon White dwarves?

 

[PeterDonis]

seeing as red giants have a lower Polytrope value than main-sequence stars, wouldn't this indicate that the Giants still have a lower GBE than the Sun and Carbon White dwarves?

Lower as a fraction of their total energy (mass), yes. Not necessarily lower in absolute terms, since that will depend on the total mass of the star.

 

[Assaltwaffle]

Lower as a fraction of their total energy (mass), yes. Not necessarily lower in absolute terms, since that will depend on the total mass of the star.

Well, once again, thank you so much for your help. My community will be very happy to get this information.

 

[PeterDonis]

You're welcome!

 

[Buzz Bloom]

We find this with GBE

Is GPE the same as GBE?

Hi Assaltwaffle:

I would much appreciate your posting the meanings of these acronyms: GPE, GBE. I tried to find useful definitions by searching acronym sources on the Internet, but I failed to find anything useful.

Regards,
Buzz

 

[Drakkith]

I would much appreciate your posting the meanings of these acronyms: GPE, GBE. I tried to find useful definitions by searching acronym sources on the Internet, but I failed to find anything useful.

GPE: Gravitational potential energy.
GBE: Gravitational binding energy.